The Southwest Center for Arithmetic Geometry

Arizona Winter School 2000
Lauter-Schoof-Zieve Course Description

The Number of Points on a Curve over a Finite Field

Lectures:

  1. Lauter: Introduction to curves over finite fields with many rational points
  2. Zieve: Curves with many points II
  3. Schoof: Curves over F2 with many points
  4. Lauter: Class field theory constructions of curves with many points
  5. Zieve: Curves of large genus with many points
  6. Schoof: Class field towers

Prerequisites:

The prerequisite for our talks is an understanding of basic properties of curves and their function fields. A convenient reference is the book `Algebraic Function Fields and Codes' by Henning Stichtenoth; below we give specific references from this book.

  • Definitions: function field, place, valuation, zero/pole, degree (Section I.1)
  • Example of rational function field (Section I.2)
  • Some understanding of the genus of a function field and of Riemann-Roch: it would be enough to understand the statement of Thm.I.5.17 on p.29 (after looking up the definitions of degree and dimension, one could define the genus to be the unique integer `g' for which this theorem is true)
  • Behavior of places in extensions of function fields -- SUM e_i f_i = n (read enough of Section III.1 to understand the statements of III.1.6, III.1.7, and III.1.11)
  • Riemann-Hurwitz-Zeuthen genus formula (Thm.III.4.12, Def.III.4.3) and basic properties of the different (III.4.11, III.5.1), notions of wild/tame ramification (III.5.4,III.5.5,III.5.7)
  • Basic properties of Galois extensions (III.7.1,III.7.2), decomposition and inertia groups (III.8.1--III.8.4)
  • Correspondence between curves and function fields (Appendix B.9,B.10,B.12)

Suggested reading:

René Schoof has provided a preliminary version of notes on "Algebraic curves and coding theory" (in dvi, ps, and pdf formats). He emphasizes that they are preliminary and not for distribution.